Cut The Knot!

An interactive column using Java applets
by Alex Bogomolny

Napoleon's Theorem

March 1999

A remarkable theorem has been attributed to Napoleon Bonaparte although his relation to the theorem is questioned in all sources available to me that mention his name. This can be said though that mathematics flourished in post-revolutionary France and mathematicians were held in great esteem in the new Empire. Laplace was a Minister of the Interior under Napoleon, albeit for short six weeks.

On the sides of a triangle construct equilateral triangles (outer or inner Napoleon triangles). Napoleon's theorem states that the centers of the three outer Napoleon triangles form another equilateral triangle. The statement also holds for the three inner triangles.

The theorem admits a series of generalizations. The add-on triangles may have an arbitrary shape provided they are similar and properly oriented. Then any triple of the corresponding (in the sense of the similarity) points form a triangle of the same shape. Another generalization was kindly brought to my attention by Steve Gray. This time, the construction starts with an arbitrary n-gon (thought to be oriented) and proceeds in (n - 2) steps. The end result at every step is another n-gon, the last of which is either regular or star-shaped. Napoleon's theorem (both for outer and inner constructions) follows when n = 3.

I shall follow the articles by B.H.Neumann (1942) and J.Douglas (1940). (K. Petr published a proof of the result discussed below in 1905. The theorem's most recent designation is PDN, Petr-Douglas-Newmann, giving credit to all three authors.)

Let the vertices of an n-gon be ordered A1, A2, ..., An and considered as complex numbers. We shall apply a certain operation to every pair of consecutive vertices.

For any two complex numbers A and B, the all-important operation is

(1)

C = (1 - c)A + cB

If c is real, points C lie on a straight line through A and B. For complex c, A,B, and C form a triangle similar to the triangle formed by points 0, 1, and c. The orientation is preserved. If c = λ + iμ, then to construct C, draw a line through (1 - λ)A + λB perpendicular to AB and mark a point at the distance μ|B - A| from AB.

When applied to all pairs of successive vertices of an n-gon P, operation (1) yields another polygon, Pc. In geometric terms, Pc consists of the apexes of similar triangles erected on the sides of the polygon P. Vertices of Pc are generated successively from those of P. Jesse Douglas calls (1) a linear polygonal transformation - LPT. LPT has several important properties. First, P and Pc are concentric, i.e. share the centroid.

(2)

Pi = (Pc)i

To obtain other properties, arrange vertices of P and Pc as n-dimensional vectors. (1) gets then represented as a linear transformation with an n×n circulant matrix Lc with the first row given by {(1 - c) c 0 ...0}. A general circulant matrix M can also be defined by its first row { 0  1 ... n-1}. Let K denote the simplest circulant {0 1 0 ... 0}. Then Kn = I (a unit matrix) and

M = 0I + 1K + 2K2 + ... + n-1Kn-1

or M = pM(K). pM(t) is known as the auxiliary polynomial of M. When two circulants are multiplied, their auxiliary polynomials are multiplied modulo Kn - I. This shows that any two circulant matrices commute, and so are LPTs. From here we also obtain a fact that is not quite obvious from geometric considerations, viz., the result of a sequence of LPTs does not depend on the order of individual transformations.

The auxiliary polynomial of a product of several LPTs is simply a product of the terms ((1 - c) + ct). For the generalization of Napoleon's theorem we are going to select LPTs in a particular manner.

Let ωd denote (n - 1) distinct roots of unity ωn = 1, excluding 1. In other words, ωd's are all roots of the equation

(3)

r(t) = tn-1 + tn-2 + ... + t + 1 = 0

For d = 1, 2, ..., n-1, define

(4)

cd = 1/(1 - ωd)

Verify that cd = (1 + i·cot(πd/n))/2 and also (1 - cd) + tcd = (t - ωd)/(1 - ωd).

With thus selected cd's, (1) is equivalent to a construction of isosceles triangles with simple apex angles. (Note that for n = 3, the angles are ±120°). From (3) and (4) we have

((1 - c1) + c1t)...((1 - cn-1) + cn-1t) = r(t)/r(1) = r(t)/n

which corresponds to the circulant matrix {1/n 1/n 1/n... 1/n}. This matrix transforms any vector-polygon A1...An into the centroid of points A1, ..., An.

Omit now any of the constituent LPTs. Let it be (1) with cd for some d. (n - 2) LPTs remain. Since LPTs commute, the polygon obtained after (n - 2) steps has the property that a single application of (1) with cd shrinks it to a point. Which exactly means that the polygon is either regular or star-shaped.

For n = 3, we get the inner or outer Napoleon triangles depending on whether c1 or c2 is omitted.


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The last stage polygons are either regular when we omit c1 or star-shaped. When n is not prime, we sometimes see a polygon with fewer sides. For example, when we start with a hexagon, the final shape may be a segment, a triangle, or another hexagon. The situation is more or less obvious. If we resort to vector/matrix notations, we may think of a polygon as the shape traced while coordinates of a vector are traversed circularly in the natural order. When n and d are mutually prime we always have n distinct points. When they are not, we are sure to have coordinates split into groups of gcd(n, d) each such that in each group coordinates are equal. Corresponding vertices of the polygon overlap.

There is a notation for regular n-gons - {n} - that also includes star-shaped ones: {n/d}. Branco Grünbaum makes an interesting point about this notation and the corresponding polygon. For simplicity, I shall assume all n-gons below have their vertices on a fixed circle.

For an integer n, sides of a regular n-gon span just 1 arc into which the circle is split by the n vertices. This explains why sometimes a regular n-gon is also called an {n/1} polygon. The polygon whose sides span d arcs at a time is denoted {n/d}.

Usuallly a (generalized) regular polygon {n/d} is defined as a shape obtained by connecting vertices of a regular n-gon while skipping d arcs at a time. The definition requires that n and d be mutually prime (e.g., [Coxeter]).

Branco Grünbaum argues that since the definition works as well for n and d that are not coprime, there is no good reason to exclude such cases. He traces the common usage to a work by Louis Poinsot (1810) who looked at {14/2} as a pair of overlapping heptagons. When we either use the definition, or simply place coordinates of an n-vector onto a circle, the point goes around the center of the circle d times, regardless of whether n and d are coprime or not. However, this argument would work for two heptagons as well as for a single polygon of type {14/2}. To prove his point Grünbaum continuously metamorphoses "nonconventional" polygons (like {14/2} that appears as a heptagon) into "conventional" ones (like {14/5}). Says he: "if polygons with coinciding vertices were banned from the discourse, our continuous families of polygons would shatter into many fragments."

The manner in which the {n/d} polygons are obtained in the generalized Napoleon theorem seems to support his point. Indeed, it seems impossible to obtain n-gons with different n's by means of LPTs. A polygon of type {6/2} is an image of a polygon of type {6} in a 6-dimensional space by a 6-dimensional transformation.

In the notation {n/d}, the slash has been looked at as division. In the Introduction to Geometry, Coxeter simply defines n/d-gons for a rational n/d. Naturally, then {14/2} is the same as {7/1}, or {7}. This is why the constructive definition needs n and d mutually prime. If the nonconvential polygons are allowed, it seems that the overused {n,d} may be more appropriate.

Reference

  1. H. S. M.Coxeter, Introduction to Geometry, John Wiley & Sons, NY, 1961
  2. J. Douglas, On Linear Polygon Transformation, Bull Amer Math Soc, 46 (1940), 551-560
  3. S. B. Gray, Generalizing the Petr-Douglas-Neumann Theorem on N-Gons, The American Mathematical Monthly, Vol. 110, No. 3 (Mar., 2003), pp. 210-227
  4. B. Grünbaum, Metamorphosis of Polygons, in The Lighter Side of Mathematics, R.K.Guy and R.E.Woodrow (eds), MAA, 1994.
  5. B. H. Neumann, A Remark on Polygons, J London Math Soc, 17 (1942), 165-166

Napoleon's Theorem

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